Tangents and radii

Objective. Use perpendicular radii and equal tangent lengths.

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Learn: the key idea

Tangents and radii properties state that a tangent to a circle is always perpendicular to the radius drawn to the point of contact, and that two tangents drawn from the same external point to a circle are equal in length. Use the perpendicular property to form right-angled triangles involving the radius, tangent and the line to the external point, often solved with Pythagoras. If a tangent length is 12 cm and the radius is 5 cm, the distance from the external point to the centre is √(12² + 5²) = 13 cm.

Worked example

A tangent from external point T touches a circle of radius 9 cm at point A, where TA = 12 cm. Use the tangents and radii relationship to find the distance from T to the centre O.

1 $Recall the radius is perpendicular to the tangent at the point of contact$
2 $This forms a right-angled triangle OAT with the right angle at A$
3 $Apply Pythagoras: OT² = OA² + AT² = 9² + 12²$
4 $Calculate: OT² = 81 + 144 = 225, so OT = 15 cm$

Answer: 15 cm

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Common mistake. Students sometimes assume the tangent is perpendicular to the line joining the external point to the centre, rather than perpendicular to the radius at the point of contact. Always mark the right angle precisely at the point where the tangent touches the circle.

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